Motivated by an Arens regularity problem, we introduce the concepts of matrix Banach space and matrix Banach algebra. The notion of matrix normed space in the sense of Ruan is a special case of our matrix normed system. A matrix Banach algebra is a matrix Banach space with a completely contractive multiplication. We study the structure of matrix Banach spaces and matrix Banach algebras. Then we...

In this paper, we introduce a new geometric property (A2) ∗ and we show that if a separable Banach space has property (A2) ∗ then both X and its dual X∗ have the weak fixed point property. Criteria for Orlicz spaces to have the properties (A2), (A ε 2) ∗ and (NUS∗) are given.

A Banach space X is said to satisfy the weak fixed point property (fpp) if every nonempty weakly compact convex subsetC, and every nonexpansivemapping T : C→ C (i.e., ‖Tx− Ty‖ ≤ ‖x− y‖ for every x, y ∈ C) has a fixed point, that is, there exists x ∈ C such that T(x) = x. Many properties have been shown to imply fpp. The most recent one is the uniform nonsquareness which is proved by Mazcuñán [2...

Given a property of Banach spaces that is hereditary, it is natural to ask whether a Banach space has the property if each of its subspaces with a w Ž . Ž . particular structure such as a Schauder basis or a Schauder finite-dix mensional decomposition has the property. The motivation for such questions is that it is much easier to deal with Banach spaces with such an additional structure. A sub...

In this paper we introduce the notion of $varphi$-commutativity for a Banach algebra $A$, where $varphi$ is a continuous homomorphism on $A$ and study the concept of $varphi$-weak amenability for $varphi$-commutative Banach algebras. We give an example to show that the class of $varphi$-weakly amenable Banach algebras is larger than that of weakly amenable commutative Banach algebras. We charac...

motivated by an arens regularity problem, we introduce the concepts of matrix banach space and matrix banach algebra. the notion of matrix normed space in the sense of ruan is a special case of our matrix normed system. a matrix banach algebra is a matrix banach space with a completely contractive multiplication. we study the structure of matrix banach spaces and matrix banach algebras. then we...

Introduced by Vitali(1907), Hahn(1922) and Saks(1933), the VitaliHahn-Saks theorem plays a vital role in measure theory in proving some weak convergence theorems. In this Toolbox Essay, we will first introduce the Nikodym theorem followed by a self-contained proof. Next, we state the Vitali-Hahn-Saks theorem and show by an example how this theorem can be used in probability theory. The benefit ...